The University of Iowa says Nagel died Jan. 15 in San Antonio. His daughter Nancy Nagel confirmed her father's death when reached by phone Wednesday.
The Los Angeles native played quarterback for UCLA and began his coaching career there as an assistant.
Nagel coached and played for the NFL's Chicago Cardinals in 1953 and worked as an assistant coach at Oklahoma from 1954 to 1957.
He was head coach at Utah from 1958 to 1965 and at Iowa from 1966 to 1970.
Nagel went on to serve as athletic director at Washington State and Hawaii, executive vice president of the Los Angeles Rams and executive director of the Hula Bowl before retiring in 1995.
He was inducted into the National Association of Collegiate Directors of Athletics Hall of Fame in 2006.